On the inversion of the block double-structured and of the triple-structured Toeplitz matrices and on the corresponding reflection coefficients

TL;DR

This paper extends classical inversion methods of Toeplitz matrices to complex block and multi-dimensional cases, providing new insights into their structure and reflection coefficients.

Contribution

It develops a novel approach for inverting block double-structured and triple-structured Toeplitz matrices, generalizing previous 2-D results to higher dimensions.

Findings

Derived inversion formulas for block double-structured Toeplitz matrices.

Extended inversion techniques to 3-D Toeplitz matrices.

Characterized reflection coefficients for complex structured matrices.

Abstract

The results on the inversion of convolution operators as well as Toeplitz (and block Toeplitz) matrices in the $1$-D (one-dimensional) case are classical and have numerous applications. Last year, we considered the $2$-D case of Toeplitz-block Toeplitz (TBT) matrices, described a minimal information, which is necessary to recover the inverse matrices, and gave a complete characterisation of the inverse matrices. Now, we develop our approach for the more complicated cases of block TBT-matrices and $3$-D Toeplitz matrices.